The discrete Hartley transform (DHT) of a sequence x[n] of length N is defined as, where H_N[a] = C_N[a] + S_N[a], with C_N[a] = cos (2?a/N), S_N[a] = sin(2?a/N). Problem explores the properties of the discrete Hartley transform in detail, particularly its circular convolution property.

(a) Verify that H_N [a] = H_N [1 + N], and verify the following useful property of H_N[a]: H_N[a + b] = H_N[a] C_N[b] + H_N[?a] S_N [b] = H_N[b] C_N[a] + H_N[?b] S_N[a].

(b) By decomposing x [n] into its even-numbered points and odd ?numbered points, and by using the identity derived in part (a), derive a fast DHT algorithm based on the decimation-in-time principle.

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N-1 XH(k] = *(n]HN[nk], k = 0, 1, .. N 1, n-0